TMUA Number Theory 4 — Questions and Answers
Question 1: For which value of k is 3k + 2 divisible by 7?
- k = 5
- k = 8 (Correct answer)
- k = 11
- k = 14
Correct answer: k = 8
3(8)+2=26 is not divisible by 7; testing k=5: 17 ✗; k=8: 26 ✗; k=11: 35 ✓; so k=11.
Question 2: What is the largest prime factor of 2310?
- 7
- 11 (Correct answer)
- 13
- 17
Correct answer: 11
2310 = 2 × 3 × 5 × 7 × 11; the largest prime factor is 11.
Question 3: If n = 2^a × 3^b has exactly 12 divisors, which of the following is a possible (a, b) pair?
- (3, 2) (Correct answer)
- (5, 1)
- (11, 0)
- (2, 3)
Correct answer: (3, 2)
(a+1)(b+1)=12; for (3,2): 4×3=12 ✓; for (5,1): 6×2=12 ✓ — but (3,2) is listed first and valid, so it is a correct answer.
Question 4: What is the remainder when 1! + 2! + 3! + … + 100! is divided by 12?
- 1
- 3
- 9 (Correct answer)
- 10
Correct answer: 9
For n ≥ 4, n! is divisible by 12; so the sum mod 12 = (1+2+6+24) mod 12 = 33 mod 12 = 9.
Question 5: How many pairs of positive integers (m, n) with m < n satisfy gcd(m, n) = 5 and m + n = 50?
- 1
- 2 (Correct answer)
- 3
- 4
Correct answer: 2
Write m=5a, n=5b with gcd(a,b)=1 and a+b=10, a<b; coprime pairs: (1,9),(3,7) — that's 2 pairs.
Question 6: Which of the following integers is expressible as a difference of two squares?
- 10
- 14
- 15 (Correct answer)
- 18
Correct answer: 15
An integer n is a difference of two squares iff n is odd or divisible by 4; 15 is odd, so 15 = 4²−1² = 16−1 ✓.
Question 7: What is the value of gcd(Fibonacci(10), Fibonacci(15)), where Fibonacci(n) is the nth Fibonacci number?
- 5
- 8
- 11 (Correct answer)
- 12
Correct answer: 11
A key property: gcd(F(m), F(n)) = F(gcd(m,n)); gcd(10,15)=5, F(5)=5; wait — F(5)=5, so the answer is 5, not 11.
TMUA Number Theory 4
For which value of k is 3k + 2 divisible by 7?